Ultraconnected space
Definition
A topological space is termed an ultraconnected space if it satisfies the following equivalent conditions:
- It is nonempty and cannot be expressed as a union of two proper open subsets
- It is nonempty and cannot be expressed as a union of finitely many proper open subsets
- It is nonempty and any two nonempty closed subsets have nonempty intersection
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| path-connected space | ultraconnected implies path-connected | |FULL LIST, MORE INFO | ||
| connected space | |FULL LIST, MORE INFO | |||
| normal space | ultraconnected implies normal | |FULL LIST, MORE INFO | ||
| pseudocompact space | |FULL LIST, MORE INFO | |||
| limit point-compact space | |FULL LIST, MORE INFO |
Opposite properties
Similar properties
- Irreducible space, with a similar definition but the roles of "open" and "closed" interchanged